The Stacks project

Lemma 64.16.3. Let $\mathit{Sch}_{fppf}$ be a big fppf site. Let $S \to S'$ be a morphism of this site. The construction above give an equivalence of categories

\[ \left\{ \begin{matrix} \text{category of algebraic} \\ \text{spaces over }S \end{matrix} \right\} \leftrightarrow \left\{ \begin{matrix} \text{category of pairs }(F', F' \to S)\text{ consisting} \\ \text{of an algebraic space }F'\text{ over }S'\text{ and a} \\ \text{morphism }F' \to S\text{ of algebraic spaces over }S' \end{matrix} \right\} \]

Proof. Let $F$ be an algebraic space over $S$. The functor from left to right assigns the pair $(j_!F, j_!F \to S)$ ot $F$ which is an object of the right hand side by Lemma 64.16.1. Since this defines an equivalence of categories of sheaves by Sites, Lemma 7.25.4 to finish the proof it suffices to show: if $F$ is a sheaf and $j_!F$ is an algebraic space, then $F$ is an algebraic space. To do this, write $j_!F = U'/R'$ as in Lemma 64.9.1 with $U', R' \in \mathop{\mathrm{Ob}}\nolimits ((\mathit{Sch}/S')_{fppf})$. Then the compositions $U' \to j_!F \to S$ and $R' \to j_!F \to S$ are morphisms of schemes over $S'$. Denote $U, R$ the corresponding objects of $(\mathit{Sch}/S)_{fppf}$. The two morphisms $R' \to U'$ are morphisms over $S$ and hence correspond to morphisms $R \to U$. Since these are simply the same morphisms (but viewed over $S$) we see that we get an ├ętale equivalence relation over $S$. As $j_!$ defines an equivalence of categories of sheaves (see reference above) we see that $F = U/R$ and by Theorem 64.10.5 we see that $F$ is an algebraic space. $\square$

Comments (0)

Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 04SG. Beware of the difference between the letter 'O' and the digit '0'.